BTW, I never understood what you meant by oscillating cylinder. Is this oscillating rotationally or up and down? Navier-Stokes could handle the first, but you need CFD for the second! (Hint: Navier-Stokes is a LOT easier in 2D)
You need to think carefully about boundary conditions too.
If you are asking how a rotating cylinder can cause a steady vortex formation - that's easy: centrifugal force...
Mart